Question

Let $s_1,s_2,s_3$ be the respective sums of $n,2n,3n$ terms of the same arithmetic progression with a as the first term and d as the common difference.
$R=$ $s_3-s_2-s_1$. Then $R$ is dependent on:

• A. $a$ and $d$
• B. $d$ and $n$
• C. $a$ and $n$
• D. $a,d$ and $n$
• E. Neither a nor d nor n

Answer 1

Answer: (B)

$d$ and $n$

$S_1=\frac{n}{2}[2a+(n-1\left)d\right],s_2=\frac{2n}{2}[2a+(2n-1\left)d\right],s_3=\frac{3n}{2}[2a+(3a-1\left)d\right].$

R = $s_3-s_2-s_1=\frac{n}{2}[6a+9nd-3d-4a-4nd+2d-2a-nd+d]=2n^{2}d,$

So that b is the correct choice.